semimetric

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semimetric

[¦sem·i′me·trik]
(mathematics)
A real valued function d (x,y) on pairs of points from a topological space which has all the same properties as a metric save that d (x,y) may be zero even if x and y are distinct points. Also known as pseudometric.
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3-8]) and, more recently, the case when the regressor takes values in a semimetric space of infinite dimension has been addressed.
Let us recall that a semimetric space (X, d), also often referred to as apseudometric space, is defined exactly like a metric space, except that the condition d(x,y) = 0 for a pair of points x,y [member of] X does not imply that x = y.
Here are three definitions and three propositions concerning an arbitrary semimetric space (X, d).
The semimetric space (X, d) is said to be totally bounded if, for each r > 0, there exists a finite set [F.
The following lemma is a corollary of Blumenthal's solution of the problem of isometric embedding of semimetric spaces in the Euclidean spaces, see [[6], p.
Despite the seeming impossibility of joining metric and non-metric properties in "one package", Smarandache semimetric spaces can easily be introduced even by means of "classical" General Relativity.